Rotational Motion

JEE Main level, single correct, medium

262 of 280

Question 262 of 280

The following bodies are made to roll up (without slipping) the same inclined plane from a horizontal plane: (i) a ring of radius R,R, (ii) a solid cylinder of radius R2\dfrac{R}{2} and (iii) a solid sphere of radius R4.\dfrac{R}{4}. If, in each case, the speed of the center of mass at the bottom of the incline is same, the ratio of the maximum heights they climb is:

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Answer: (B) 20:15:1420:15:14

From conservation of mechanical energy.
(12mRv2+12IRωr2)=mRghR⇒V22+V22=ghR\left(\dfrac{1}{2}{m}_{R}{v}^{2}+\dfrac{1}{2}{I}_{R}{\omega }_{r}^{2}\right)={m}_{R}g{h}_{R}\Rightarrow \dfrac{{V}^{2}}{2}+\dfrac{{V}^{2}}{2}=g{h}_{R}
(12mdv2+12Idωd2)=mdghd⇒V22+V24=ghd\left(\dfrac{1}{2}{m}_{d}{v}^{2}+\dfrac{1}{2}{I}_{d}{\omega }_{d}^{2}\right)={m}_{d}g{h}_{d}\Rightarrow \dfrac{{V}^{2}}{2}+\dfrac{{V}^{2}}{4}=g{h}_{d}
(12msv2+12Isωs2)=msghs⇒V22+V25=ghs\left(\dfrac{1}{2}{m}_{s}{v}^{2}+\dfrac{1}{2}{I}_{s}{\omega }_{s}^{2}\right)={m}_{s}g{h}_{s}\Rightarrow \dfrac{{V}^{2}}{2}+\dfrac{{V}^{2}}{5}=g{h}_{s}
∴hR:hd:hs=1:34:710=20:15:14∴{h}_{R}:{h}_{d}:{h}_{s}=1:\dfrac{3}{4}:\dfrac{7}{10}=20:15:14

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